# Prime signature

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The prime signature of a number is the multiset of exponents of its prime factorisation. The prime signature of a number having prime factorisation ${\displaystyle p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{n}^{m_{n}}}$ is ${\displaystyle \left\{m_{1},m_{2},\dots ,m_{n}\right\}}$.

For example, all prime numbers have a prime signature of {1}, the squares of primes have a prime signature of {2}, the products of 2 distinct primes have a prime signature of {1,1} and the products of a square of a prime and a different prime (e.g. 12,18,20,... ) have a prime signature of {2,1}.

The number of divisors that a number has is determined by its prime signature as follows: If you add one to each exponent and multiply them together you get the number of divisors including the number itself and 1. For example, 20 has prime signature {2,1} and so the number of divisors is 3 × 2 = 6. They are 1, 2, 4, 5, 10 and 20.

The smallest number of each prime signature is a product of primorials. The first few are:

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, ... (sequence A025487 in the OEIS).

The divisor function τ(n), the Möbius function μ(n), the number of distinct prime divisors ω(n) of n, the number of prime divisors Ω(n) of n, the indicator function of the squarefree integers, and many other important functions in number theory, are functions of the prime signature of n.

## Numbers with same prime signature

Signature Numbers OEIS ID Description
1 The number 1, as an empty product of primes
{1} 2, 3, 5, 7, 11, ... A000040 prime numbers
{2} 4, 9, 25, 49, 121, ... A001248 squares of prime numbers
{1,1} 6, 10, 14, 15, 21, ... A006881 two distinct prime divisors (square-free semiprimes)
{3} 8, 27, 125, 343, ... A030078 cubes of prime numbers
{2,1} 12, 18, 20, 28, ... A054753 squares of primes times another prime
{4} 16, 81, 625, 2401, ... A030514 fourth powers of prime numbers
{3,1} 24, 40, 54, 56, ... A065036 cubes of primes times another prime
{1,1,1} 30, 42, 66, 70, ... A007304 three distinct prime divisors (sphenic numbers)
{5} 32, 243, 3125, ... A050997 fifth powers of primes
{2,2} 36, 100, 196, 225, ... A085986 squares of square-free semiprimes

## Sequences defined by their prime signature

Given a number with prime signature S, it is